In this paper several results concerning the periodic points of 1-norm non-expansive maps will be presented. In particular, we will examine the set $R(n)$, which consists of integers $p\geq 1$ such that there exist a 1-norm nonexpansive map $f{:}\ \mathbb{R}^n\rightarrow\mathbb{R}^n$ and a periodic point of $f$ of minimal period $p$. The principal problem is to find a characterization of the set $R(n)$ in terms of arithmetical and combinatorial constraints. This problem was posed in [12, section 4]. We shall present here a significant step towards such a characterization. In fact, we shall introduce for each $n\in\mathbb{N}$ a set $T(n)$ that is determined by arithmetical and combinatorial constraints only, and prove that $R(n)\subset T(n)$ for all $n\in \mathbb{N}$. Moreover, we will see that $R(n)=T(n)$ for $n=1,2,3,4,6,7$, and 10, but it remains an open problem whether the sets $R(n)$ and $T(n)$ are equal for all $n\in \mathbb{N}$.